Westergaard theory; warping and friction stresses

Westergaard theory: assumptions and analysis

Westergaard (1926) analysed a concrete slab resting on an elastic subgrade, treating the slab as a thin plate on a liquid (Winkler) foundation.

Assumptions

1. The slab acts as a homogeneous, isotropic elastic plate of uniform thickness.
2. The reaction of the subgrade is vertical and proportional to deflection ($p=k\,\Delta$), where $k$ is the modulus of subgrade reaction (a "dense liquid"; no shear transfer in the subgrade).
3. The slab is in full contact with the subgrade at all times.
4. The load is applied on a circular area of radius $a$ with uniform pressure; the thickness is small compared with the other dimensions and the slab is treated as infinite or semi-infinite for the three cases.
5. The load is static; temperature, joint load transfer and friction are not included in the basic analysis.

Analysis - three loading positions

• Interior load (far from edges):

$$\sigma_i=\frac{0.316\,P}{h^2}\left[4\log_{10}\frac{l}{b}+1.069\right]$$

• Edge load (at the free edge):

$$\sigma_e=\frac{0.572\,P}{h^2}\left[4\log_{10}\frac{l}{b}+0.359\right]$$

Teller and Sutherland's modification (the basis of the IRC edge chart) is $\sigma_e=\dfrac{0.529P}{h^2}(1+0.54\mu)\left[4\log_{10}\dfrac{l}{b}+\log_{10}b-0.4048\right]$.

• Corner load:

$$\sigma_c=\frac{3P}{h^2}\left[1-\left(\frac{a\sqrt2}{l}\right)^{0.6}\right]$$

Kelly's modification (used by IRC) changes the exponent from 0.6 to 1.2.

Here $P$ is the wheel load, $h$ thickness, $a$ contact radius, $b=\sqrt{1.6a^2+h^2}-0.675h$ for $a<1.724h$ (else $b=a$) and $l=\left[\dfrac{Eh^3}{12(1-\mu^2)k}\right]^{1/4}$ the radius of relative stiffness. For the same load the edge stress is the largest bending stress and the interior stress is the smallest; the corner load gives the largest deflection.

Development of warping and friction stresses

Warping stress

Warping stress comes from the temperature difference between the top and bottom of the slab.

• By day, the top is hotter than the bottom, so the top tries to expand more and the slab edges curl downwards. The weight of the slab and the subgrade resist this, so compression develops at the top and tension at the bottom.
• At night the top is cooler, the edges curl upwards and tension develops at the top.

Bradbury gave the edge stress for a finite slab:

$$\sigma_{te}=\frac{C\,E\,\alpha\,\Delta T}{2}$$

and for the interior $\sigma_{ti}=\dfrac{E\alpha\Delta T}{2}\left(\dfrac{C_x+\mu C_y}{1-\mu^2}\right)$. The coefficient $C$ depends on $L/l$ and $B/l$ (Bradbury chart), $E$ is the modulus of concrete, $\alpha$ the thermal coefficient and $\Delta T$ the top-bottom temperature difference.

Frictional stress

When the whole slab expands or contracts with an average temperature change, the movement is resisted by friction between the slab and the sub-base. At the centre of the slab the tension from contraction is

$$\sigma_f=\frac{W\,L\,f}{2\times10^{4}}\ \ (\text{kg/cm}^2)$$

with $W$ the unit weight of concrete (2400 kg/m3), $L$ the joint spacing (m) and $f$ the coefficient of friction (about 1.5). It is tensile in winter or at night (contraction) and compressive in summer. It is generally small and fixes the spacing of contraction joints.